key words: rate, ratio, per, proportion, exchange, yield, interest

Key words: rate, ratio, per, proportion, exchange, yield, interest.

Purpose: This unit is designed to help tutors who teach courses in which working with rates is important. These courses may involve, rates of exchange, interest rates, yield rates, time and distance rates, rates of coverage

Tutor Outcomes: By the end of the unit tutors should be able to: 1. Recognise contexts and problems that involve rates and can be solved using proportional thinking2. Develop lessons in their teaching context that help learners to solve problems using rates.

Resources you will need for this unit:Calculators

Overview

Mathematical Background

A unit rate is essentially a ratio used to compare a quantity (unit) to the way it is being measured. The two numbers being compared are not the same type, they are different units e.g dollars per kilogram, NZ Dollars per US Dollars

A rate describes a relationship between two different measures. (Some rates have more than 2 units, eg., an IV rate might be express mg/kg/hour)In describing the units of a rate, the word "per" is used to separate the units of the two measurements used to calculate the rate (for example a heart rate is expressed "beats per minute").

The two parts are not mixed.Rate example: 50km/hr, comparison of speed to time.Ratio example: 1:2 fertiliser to water (the water and the fertiliser can be mixed)

http://en.wikipedia.org/wiki/Heart_rate


Why do we work with rates?•The basis of calculating an amount or value as a reference point. •A benchmark for calculating more or less, bigger or smaller, faster or slower.•To keep the rates constant we need to keep the amounts in proportion.

Rates have two features•Two measurement units, e.g for the price of coffee the units were dollars and grams•The word “per” which means “for every”, the cost of coffee was $3.50 for every 100 grams ($3.50 per 100g).

Recognising Rates

Some contexts are easy to recognise as rates:

60 km/hr,

$5 for 50 Mb data,

$14 / hr

1 litre per 15 m2

Examples

Examples

Recognising Rates

Some contexts are less easy to recognise as rates:

50 mg tablet

3 m length boards

60% yield

It takes 2 sheets of metal to make 3 ducts

Examples

Working with rates

Rates are often expressed using the symbol / to mean ‘per’ or ‘for every’. The same symbol is used for division in some contexts. The symbols used in maths often get in the way of understanding for learners who struggle with maths.

A good teaching tip for working with rates and proportions is to organise what we know, and what we’re looking for in a table or box.

For example 80 km/hr involves two units, km and hours:

Km 80

hours 1

Examples

Working with rates

Start with a context most learners can relate to.

Example: Sam’s average speed for a road trip is 80 km/hr. How long will it take him to travel 320 km?

Think about what 80 km/hr means…

Every 1 hour Sam travels 80 km. So in 2 hours he travels…

Km 80 ?

hours 1 2

Examples

Working with rates


Now build up the table:

Every 1 hour Sam travels 80 km. So in 2 hours he travels 160 km. How many hours to go 320 km?

Km 80 160 320

hours 1 2 ?

x 2

x 2

x 2

x 2


Direct Proportionality

To keep the rate constant we need to keep the amounts in direct proportion. If the value of one unit in a rate doubles then the value of the other unit doubles, and so on.

Our table so far, gives us three ways of expressing a constant rate: 80 km/hr is the same rate as 160 km/ 2hr or 320 km/ 4hr.

Note there are horizontal and vertical relationships between the numbers.

Km 80 160 320

hours 1 2 4

x 4

x 4

÷ 80 ÷ 80


Proportionality

There is also a relationship between diagonal numbers:

80 x 2 = 1 x 1602 x 320 = 160 x 41 x 320 = 80 x 4

This is a very useful check when we’re solving problems using rates.

Km 80 160 320

hours 1 2 4

Examples

Finding less obvious values


This is a slightly harder problem because 300 is not a whole number multiple of 80.

You and your learners may come up with several strategies for working out the answer. But before you even start looking, ask yourself (and your learners): What do we already know about the answer?

We already know that the answer is between 2 and 4 hours, (and it’s closer to 4 than 2.)

Km 80 160 320

hours 1 2 4

Examples



Note that we can build up the table using additive thinking:

Sam travels 80 km in 1 hour and 160 km in 2 hours so in 3 hours he travels160 + 80 = 240 km

OR multiplicative thinking:

Sam travels 80 km in 1 hour and 3 hours is 3 times longer; so in 3 hours he travels80 x 3 = 240 km

Km 80 160 240 320

hours 1 2 3 4

Examples



Because 300 km is between 240 and 320, we know it takes between 3 and 4 hours to travel 300 km. Do we need to be more exact than this? It depends on the context.

If we need an exact answer, several strategies are available.

Km 80 160 240 320

hours 1 2 3 4

Examples



Possible Strategy 1: Extend the table for km per part hours

It takes ¼ hour to travel 20 km and ½ hour to travel 40 km, so ¾ hour to travel 60 km. (Note that the table can be built up in either direction.)

3 hours takes Sam 240 km, ¾ hour will take him another 60 km. So to travel 300 km takes 3 ¾ or 3.75 hours.

This strategy will suit learners who are more comfortable thinking additively than multiplicatively.

Km 20 40 80 160 240 320

hours 0.25 0.5 1 2 3 4

Examples



Possible Strategy 2: Work out how many times bigger 300 is than 240, and apply that multiple to 3.

300 ÷ 240 = 1.25 240 x 1.25 = 300

Km 80 160 240 300 320

hours 1 2 3 ? 4

x ?

Km 80 160 240 300 320

hours 1 2 3 3.75 4

x 1.25

x 1.25

We now know that 300 is 1.25 times bigger than 240

So the travel time is 1.25 times bigger than 3 hours

Examples



Check the answer using the diagonals:

3 x 300 = 900240 x 3.75 = 900

Km 80 160 240 300 320

hours 1 2 3 3.75 4

x 1.25

x 1.25

Examples



Possible Strategy 3: Use the diagonal pattern to find the missing value

3 x 300 = 240 x ? 900 = 240 x ? ? = 900 ÷ 240

Km 80 160 240 300 320

hours 1 2 3 ? 4

Km 80 160 240 300 320

hours 1 2 3 1.25 4

We know that 3 x 300 = 240 x missing value

Note that to use this strategy with understanding, your learners must understand that 900 ÷ 240 means 240 x ‘what’ = 900

Examples



The beauty of the diagonals is that it doesn’t matter which diagonal combination you use:

1 x 300 = 80 x ? 300 = 80 x ? ? = 300 ÷ 80

Km 80 160 240 300 320

hours 1 2 3 ? 4

Km 80 160 240 300 320

hours 1 2 3 1.25 4

Board lengths are 1.5 m, how many boards are needed to cover 4.5 m?

How might you solve it?Are there any problems like this in the context of your course?

Examples

Worked Examples

The implied rate in this problem is 1.5 m per board

Now build up the table:

Every 1 board covers 1.5 m. So 2 boards cover 3 m. How many boards will cover 4.5 m?

metres 1.5 3 4.5

boards 1 2 ?

x 2

x 2


Worked Examples

Additive Strategy: 1 board covers 1.5 m and 2 boards cover 3 m. So 3 boards cover 1.5 + 3 = 4.5 m.

metres 1.5 3 4.5

boards 1 2 3


Worked Examples

Multiplicative strategy 1:

Find how many times bigger 4.5 is than 1.5 4.5 ÷ 1.5 = 3

Apply that multiple to both rows:

metres 1.5 4.5

boards 1 3


x 3

x 3

Worked Examples

Multiplicative strategy 2 (diagonals):

1 x 4.5 = 1.5 x ? ? = 1 x 4.5 ÷ 1.5 = 3

metres 1.5 4.5

boards 1 ?


OR:

2 x 4.5 = 3 x ? ? = 2 x 4.5 ÷ 3 = 3

metres 1.5 3 4.5

boards 1 2 ?

Suppose 1 NZ dollar buys 76 Australian cents:$NZ1 = $AU0.76How much Australian currency will $NZ1350 buy?

• Can you solve this using a table?• Any similar problems in your course?

Worked Examples

Worked Examples

Now build up a table:

1 NZ dollar buys Aussie $0.76. So 10 NZ dollars buys $7.60 and 100 NZ dollars buys $AU76.00. And so on…

NZ $ 1 10 100 1000

AU $ 0.76 7.60 76 760

x 10

X 10

$NZ1 = $AU0.76How much Australian currency will $NZ1350 buy?

Worked Examples

Additive strategy:

$NZ1000 buys $AU760Plus $NZ300 buys 3 x $AU76 (3 x 76 = $AU228)Plus $NZ50 buys half of $AU76 (76 ÷ 2 = $AU38)

$NZ1350 = $AU (760 + 228 + 38) = $AU 1026

NZ $ 1 10 50 100 300 1000

AU $ 0.76 7.60 38 76 228 760


How would you use this additive strategy to work out how much $NZ1357.50 would buy?

Worked Examples

Multiplicative strategy:

$NZ1 buys $AU0.76

$NZ1350 is 1350 times bigger than $NZ1 so $AU0.76 x 1350 = $1026

NZ $ 1 1350

AU $ 0.76


x 1350

x 1350

Worked Examples

We can build the table up to estimate that we’ll need a bit less than $NZ1500.

NZ $ 1 1000 500 1500

AU $ 0.76 760 380 1140

A slight twist on the problem:

$NZ1 = $AU0.76How much NZ currency is required to buy $AU1000?

Worked Examples



To work out the exact answer for this problem, it’s useful to work out how many $NZ are equivalent to $1 Aussie.

NZ $ ? 1

AU $ 1 0.76

÷ 0.76

To change 0.76 to 1, we divide by 0.76. To keep the table proportional, we need to also divide the top line by 0.76

÷ 0.76

Worked Examples



1 ÷ 0.76 = 1.315, so every 1 Aussie $ buys $NZ1.32

NZ $ 1.315 1

AU $ 1 0.76

÷ 0.76

÷ 0.76

Now build the table to $AU1000 NZ $ 1.315 1 1315

AU $ 1 0.76 1000

x 1000

x 1000

Activity

Use the rates in your course or context Think about the calculations in your course that are based on rates.

Get your learners to set the problems up using a proportion table (ratio box).

Ask them to think about what would be a reasonable answer before they calculate the exact answer.

Practice Problems

Following are some problems from various contexts to practice with.

It takes two sheets of metal make three ducts. How many ducts can we make from 12 sheets?

Sheets 2 12

Ducts 3 ?

Practice Problems

You get 4.5 glasses of wine from a bottle, how many bottles are required for 90 glasses of wine?

Glasses 4.5 90

Bottles 1 ?

You get 4.5 glasses of wine from a bottle, how many glasses of wine can you get from a case of 12 bottles?

Glasses 4.5 ?

Bottles 1 12

Practice Problems

Since percent means ‘for every 100’, percentage problems are also rate problems:

In Frank’s business last month, electricity costs were $900 and total costs were $4000. Electricity accounts for what percentage of the total expenses?

Electricity ? 900

Total costs 100 4000

key words: rate, ratio, per, proportion, exchange, yield, interest

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